9.1 Use of Baye's Theorem

Let us consider the following cases,

A group of students conducted a survey regarding smoking and lung diseases.The world has $34\%$ of its population as smokers. As per records, $48\%$ of the population has one or the other kind of lung disease. The study revealed that about $46\%$ of people having lung disease were smokers. One of the students, feels the survey should have been the other way. They should have surveyed smokers, and should have found how many smokers have the lung disease. In this way, one can tell how smoking causes lung diseases.

In this study, lets consider the following cases,

By this the students were able to make out that only $46\%$ of the people were smokers. This creates an impression that smoking is not the major cause of the disease.

In this case, fraction of people having lung disease who were smokers was collected.

i.e. the study was not sufficient enough to reveal how much smoking affects the lungs.

In case someone smokes, they can defend themselves saying that only $46\%$ of the lung diseased were smokers, and that they can't be one of them so easily.

Let us use Baye's theorem and consider the smokers as sample space.

We have the total population of people having lung disease = $48\%$

We have the fraction of smokers who have disease = $46\%$

We also have the fraction of smokers in the world = $34\%$

So, considering data to be correct, if the smokers were considered the sample space,

$${{46} \over {100}} \times {{48} \over {100}} = x \times {{34} \over {100}}$$

Here $x$ is the fraction of smokers who have lung disease.

$${{0.48 \times 0.46} \over {0.34}} = x$$

$$x = 64.9$$

By this, we are sure that around $70\%$ of smokers suffer from the disease.

Hence a large population is being affected by smokers.

Since the study was conducted on the relation, we require the smokers suffering from disease, and not the sufferers who are smokers. This way they can claim that $70\%$ of smokers are prone to have the disease. So, just in case someone smokes, it can be mentioned that they can fall in the $70\%$ more easily.

Baye's theorem is also called reverted conditional probability, where the sample space and conditions are exchanged.

For example,

Finding the probability of a person being a smoker given that he is diseased is conditional probability.

Finding the probability that a person is diseased given that he is a smoker is Baye's theorem. Using this we can give a much stronger claim on how smoking is affects lungs than the above.